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Middle Path table of five-limit rank two temperaments

Edit 9 27 …
This table is an an updated version of Table 1 of Paul Erlich's A Middle Path. The complexity is now measured a different way (which, however, is proportional to the original complexity for this table), and some temperament names have been updated.

The "main sequence" of this table comprises all possible five-limit 2-D cases where complexity/7.65 + damage/10 < 1. The 'exotemperaments' have larger damage and lend themselves to special measures such as custom inharmonic timbres in order to aid harmoniousness. The 'bonus temperaments' are more complex but have exceedingly small damage and sound like JI.

See also Middle Path table of seven-limit rank two temperaments, Middle Path table of eleven-limit rank two temperaments.

Vanishing Interval's Ratio
 V.I. Vector
 V.I. cents
 Temperament
name
 TOP
period
 TOP
generator
 Mapp.
of 2
 Mapp.
of 3
 Mapp.
of 5
 Cmplx.
 TOP
Dmg.
 ETs
TWO EXOTEMPERAMENTS:
16/15
 [4, -1, -1>
 111.7
 Father
 1185.9
 447.4
 1, 0
 2, -1
 2, 1
 1.37
 14.13
27/25
 [0, 3, -2>
 133.2
 Bug
 1200.0
 260.3
 1, 0
 2, -2
 3, -3
 1.63
 14.18
MAIN SEQUENCE:
25/24
 [-3, -1, 2>
 70.7
 Dicot
 1207.66
 353.22
 1, 0
 1, 2
 2, 1
 1.60
 7.66
 24c, 41cc, (48cc), 65ccc...
81/80
 [-4, 4, -1>
 21.5
 Meantone
 1201.70
 504.13
 1, 0
 2, -1
 4, -4
 2.19
 1.70
 12, 19, (24), 26, 29c, 31, 33c, (36)...
128/125
 [7, 0, -3>
 41.1
 Augmented
 399.02
 93.15
 3, 0
 5, -1
 7, 0
 2.42
 2.94
 12, 15, (24), 27, (30), 33, (36), 39...
135/128
 [-7, 3, 1>
 92.2
 Mavila
 1206.55
 685.03
 1, 0
 1, 1
 4, -3
 2.44
 6.55
 37bc, 67bbccc, (74bbccc), 81bbcccc...
250/243
 [1, -5, 3>
 49.2
 Porcupine
 1196.91
 1034.59
 1, 0
 -1, 3
 -2, 5
 2.75
 3.09
 15, 22, 29, (30), 36c, 37, 43c, (44)...
256/243
 [8, -5, 0>
 90.2
 Blackwood
 238.87
 158.78
 5, 0
 8, 0
 11, 1
 2.76
 5.67
 15, 25, (30), 35bc, 40b, (45bc)...
648/625
 [3, 4, -4>
 62.6
 Diminished
 299.16
 197.49
 4, 0
 7, -1
 10, -1
 3.23
 3.36
 12, (24), 28, 32c, (36), 40, 44c, (48c)...
2048/2025
 [11, -4, -2>
 19.6
 Srutal
 599.56
 494.86
 2, 0
 4, -1
 3, 2
 3.81
 0.89
 12, 22, (24), 34, (36), (44), 46, (48c), 54...
3125/3072
 [-10, -1, 5>
 29.6
 Magic
 1201.28
 380.80
 1, 0
 0, 5
 2, 1
 4.02
 1.28
 19, 22, (38), 41, (44), 54b, (57), 60...
6561/6250
 [-1, 8, -5>
 84.1
 Ripple
 1203.32
 101.99
 1, 0
 2, -5
 3, -8
 4.38
 3.32
 12, (24), (36), 47, (48c), 59b, (60c)...
15625/15552
 [-6, -5, 6>
 8.11
 Hanson
 1200.29
 317.07
 1, 0
 0, 6
 1, 5
 4.83
 0.29
 19, 34, (38), 49, 53, (57), 64b, (68)...
16875/16384
 [-14, 3, 4>
 51.1
 Negri
 1201.82
 1075.68
 1, 0
 -2, 4
 5, -3
 4.86
 1.82
 19, 29, (38), 48, (57), (58c), 66b, 67c...
20000/19683
 [5, -9, 4>
 27.7
 Tetracot
 1199.03
 176.11
 1, 0
 1, 4
 1, 9
 4.95
 0.97
 27, 34, 41, 48, (54c), 55c, 61, (68), 75...
20480/19683
 [12, -9, 1>
 68.7
 Superpyth
 1197.60
 708.17
 1, 0
 1, 1
 -3, 9
 4.95
 2.40
 22, 27, (44), 49, (54c), (66), 71, 76bc...
32805/32768
 [-15, 8, 1>
 1.95
 Helmholtz
 1200.07
 701.79
 1, 0
 1, 1
 7, -8
 5.20
 0.07
 41, 53, 65, 70c, 77, (82), 89, 94, 99c...
78732/78125
 [2, 9, -7>
 13.4
 Sensi
 1199.59
 756.60
 1, 0
 6, -7
 8, -9
 5.63
 0.41
 19, (38), 46, (57), 65, 73, (76), 84...
262144/253125
 [18, -4, -5>
 60.6
 Passion
 1198.31
 98.40
 1, 0
 2, -5
 2, 4
 6.23
 1.69
 73, 134bc, (146bc), 158bcc...
393216/390625
 [17, 1, -8>
 11.4
 Würschmidt
 1199.69
 812.05
 1, 0
 7, -8
 3, -1
 6.44
 0.31
 34, 65, (68), 96, 99, (102)...
531441/524288
 [-19, 12, 0>
 23.5
 Compton
 100.05
 15.13
 12, 0
 19, 0
 28, -1
 6.59
 0.62
 60, 72, 84, 96, 108, (120c), 132...
1600000/1594323
 [9, -13, 5>
 6.15
 Amity
 1199.85
 860.38
 1, 0
 -2, 5
 -7, 13
 7.14
 0.15
 53, 99, (106), 145, 152, (159)...
2109375/2097152
 [-21, 3, 7>
 10.1
 Orson
 1200.24
 271.65
 1, 0
 0, 7
 3, -3
 7.28
 0.24
 53, (106), 137, (159), 190...
TWO BONUS TEMPS:
6115295232/6103515625
 [23, 6, -14>
 3.34
 Vishnu
 599.97
 71.15
 2, 0
 4, -7
 5, -3
 11.27
 0.05
274877906944/274658203125
 [38, -2, -15>
 1.38
 Luna
 1199.98
 193.196
 1, 0
 4, -15
 2, 2
 13.17
 0.02

ET
 TOP damage
 TOP octave
 Rank-2 temperaments supported
12
 3.56
 1197.67
 meantone, augmented, diminished, srutal, ripple
15
 5.67
 1194.33
 augmented, porcupine, blackwood
19
 2.28
 1202.28
 meantone, magic, hanson, negri, sensi
22
 3.22
 1198.72
 porcupine, srutal, magic, superpyth
24 (contorted)
 3.56
 1197.67
 meantone, augmented, diminished, srutal, ripple
24c
 7.87
 1207.87
 dicot
25
 6.16
 1194.83
 blackwood
26
 3.69
 1203.69
 meantone
27
 2.94
 1197.06
 augmented, tetracot, superpyth
28
 5.15
 1205.15
 diminished
29
 3.47
 1202.53
 porcupine, negri
29c
 5.89
 1194.11
 meantone
30 (contorted)
 5.67
 1194.33
 augmented, porcupine, blackwood
31
 1.81
 1201.47
 meantone
32c
 5.61
 1194.39
 diminished
33
 6.43
 1200.54
 augmented
33c
 4.90
 1204.90
 meantone
34
 1.24
 1198.76
 srutal, hanson, tetracot, wuerschmidt
35bc
 9.43
 1209.43
 blackwood
36 (contorted)
 3.56
 1197.67
 meantone, augmented, diminished, srutal, ripple
36c
 4.25
 1204.25
 porcupine
37
 3.64
 1196.36
 porcupine
37bc
 6.62
 1206.62
 mavila
38 (contorted)
 2.28
 1202.28
 meantone, magic, hanson, negri, sensi

mp5r2ms_cd.png

The bubble chart above (source) arranges the temperaments by complexity (x-axis) vs. damage (y-axis). Colors are arbitrary. The size of the bubble is proportional to the size of the comma (vanishing interval). Note that the "exotemperaments" and "bonus temperaments" are not shown (a much larger range is needed to make them visible).